Collection SFN 13, 01002 (2014)DOI: 10.1051/sfn/20141301002C© Owned by the authors, published by EDP Sciences, 2014

Neutrons and magnetism

Mechthild Enderle

Institut Laue-Langevin, 6, Rue Jules Horowitz, BP. 156, 38042 Grenoble Cedex, France

Abstract. Neutron scattering is a unique technique in magnetism, since it measuresdirectly the Fourier transform of the time-dependent magnetic pair correlations. The neutroninteracts only weakly with matter, so that each neutron normally only scatters once in thesample volume. The dynamic pair correlation functions are thus probed in the whole volumeof the sample. Here we summarize the relevant formalism and approximations for neutronscattering on magnetic materials, focusing on the investigation of single crystals. The lectureaims to stimulate – not to replace – the study of the relevant literature [1–9].

1. THERMAL NEUTRON SCATTERING

1.1 Properties of the neutron

We recall some important properties of the neutron:

– wave (interference, even with itself)– particle (mass m, energy-momentum relation, interactions)– lifetime of free neutron: 15 min (allows to do experiments)– momentum p = �k = � 2�

�

– energy E = p2

2m= �

2

2mk2 = 2.072 k2 meV Å

2

– no charge– spin 1/2 and magnetic moment �n = −��N �, where � = 1.913, �N = e�

2mpis the nuclear magneton,

and � is the Pauli spin operator.

1.2 Master equation

All interactions of thermal (slow) neutrons with matter are weak, either due to the weakness of theinteraction itself (interaction with the unpaired electrons, Foldy, spin-orbit), or, as in the case of thestrong interaction with the nucleus, due to the extreme short-range character of the interaction andthe extreme “dilution” of the point-like scattering nuclei in the scattering volume. The neutron thereforeprobes the whole volume, and generally each neutron scatters only once within the sample volume.Further, the scattering process can be treated in first order perturbation, the first Born approximation

This is an Open Access article distributed under the terms of the Creative Commons Attribution License 4.0, which permitsunrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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is valid, and the differential cross section can be calculated according to Fermi’s golden rule, withthe scattering amplitude given by the transition matrix element of the interaction potential (neutronproperties are colored blue, sample properties red, and the interaction green):

d2�

d�dEf

∣∣n0,�i→n1,�f

= kf

ki

( m

2��2

)2 ∣∣〈kf �f n1|V |ki�in0〉∣∣2

�(ε1 − ε0 − (Ei − Ef ))

d2�

d�dEf

∣∣n0,�i→n1,�f

= kf

ki

( m

2��2

)2 ∑�i , �f

n0, n1

p(�i)p(n0)∣∣〈kf �f n1|V |ki�in0〉

∣∣2�(ε1 − ε0 − ��),

where

d� solid angle of the outgoing neutron beam captured by the analyzer/detector unitdEf energy interval accepted by the analyzerki , kf neutron initial and final wave vector (momentum)m neutron mass�i , �f initial and final neutron spin staten0, n1 quantum numbers for the sample’s initial and final stateε0,1 initial/final sample energyEi,f initial/final neutron energy�� = Ei − Ef energy transferV (r) the interaction potentialp(n0) probability to find the sample in the initial state n0

p(�i) probability to find the neutron in the initial spin state �i .

p(n0) = e−

εn0kBT

ne

− εnkBT

The neutron beam is generated and analyzed far from the sample, so that the incoming neutron statecan be assumed as a plane wave state eikir . The near-point-like scatterer(s) emit spherical wave(s) andare detected far away, so that their interference, the outgoing neutron wave, again can be treated asplane wave. The matrix element can therefore be rewritten as space Fourier transform of the interactionpotential, taken at the momentum transfer Q = ki − kf . (This is valid for an interaction potential thatcommutes with Q · r .) Thermal neutron energies are for most isotopes far away from nuclear resonances(with few exceptions) so that absorption often can be neglected.

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scattering length

nucleus +6.5 fmelectronic magn. mom. −5.4 fm (·S(J))electric field +1.5 am

Figure 1. Average absolute coherent nuclear scattering length in comparison to the “magnetic scattering length”�r0 of a spin 1 and to the “electric scattering length” �rp of one elementary charge. The coherent nuclear scatteringlength can have positive or negative sign and depends on the isotope, there is no systematic variation with the atomicnumber Z. r0 = �0

4�e2

meis the first Bohr radius, and rp the corresponding quantity replacing the electron mass me by

the proton mass mp .

1.3 Types of interaction

The neutron interacts with matter via different forces:

neutron – atomic nucleus strong interactionneutron – electronic magnetic moment dipole-dipole interactionneutron – electric field spin-orbit + Foldy interaction

The interaction with the electronic magnetic moment is of the same order of magnitude as the interactionwith the atomic nuclei – this is one of the reasons why neutron scattering is such a powerful techniquefor magnetism. The “scattering length” corresponding to the interaction with electric fields is about threeorders of magnitude weaker, i.e. the corresponding cross section is six orders of magnitude smaller.

1.4 Nuclear interaction potential

The Fermi pseudopotential VN for neutron scattering from one nucleus can be written as

VN (r) = 2��2

mb �(r − Rj )

b = A(I ) + B(I ) � · I

the scattering length b depends on• the isotope• the nuclear spin value I

• the total spin of neutron+nucleus: b± if the total spin is I ± 12 ·

The mean value b (average over different isotopes and nuclear spin states) for a given crystal site leadsto interference between the scattered waves from different sites and thus coherent scattering (nuclear

Bragg peaks, phonons), the standard deviation from the mean value,√

(b2) − (b)2, leads to incoherentscattering.

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1.5 Magnetic interaction potential

The magnetic potential VM for the scattering of one neutron with magnetic moment �n from themagnetic field Be(r) of one electron is

VM (r) = −�n · Be(r)

Maxwell equation: ∇ · Be(r) = 0

Contract e−ikf r and eikir to eiQr and Fourier transform the interaction

〈kf �f n1| − �n · Be(r)|ki�in0〉 = 〈�f n1| − �n · Be(Q)|�in0〉Fourier transformed Maxwell equation

Q · Be(Q) = 0.

The divergence-free magnetic field of the electron is hence always perpendicular to the momentumtransfer. In consequence, the neutron “feels” only that part of the electronic magnetic field (coming fromboth spin and orbital moment) which is perpendicular to the momentum transfer [14]. This will provideus with an easy-to-remember motivation of the so-called “magnetic selection rule”. To obtain the relationbetween magnetic moment direction and momentum transfer, we write the magnetic field Be = ∇ × A

of an electron via the vector potential A = ∇ × �tote

r, where �tot

e = �Se + �L

e . The interaction potentialbecomes

−�n · ∇× ( ∇vector potential A)

VM (r) = −�n · ∇ ×︷ ︸︸ ︷( ∇ × �S

e

r︸ ︷︷ ︸+∇ × �Le

r︸ ︷︷ ︸)

−�n · ∇× ( ∇spin orbital current)

= �n(�Se + �L

e )

r3−

3(�n · r)((

�Se + �L

e

) · r)

r5

∣∣∣∣∣︸ ︷︷ ︸−

∣∣∣∣∣8�

3�n(�S

e + �Le ) �(r).

︸ ︷︷ ︸dipole-dipole interaction contact term

The contact term describes the interaction when neutron and electron are at exactly the same place,which is possible since they are distinguishable particles.

We consider the magnetic matrix element of one electron which is proportional to

〈kf �f n1| − �n · ∇ ×(∇ × �tot

e

r

)|ki�in0〉.

The next step, “contract the plane wave states and take the Fourier transform of the interaction potential”is not obvious, derivations can be found in [1, 2, 4]. A crucial point is that VM commutes with Q · r

in spite of the explicit dependence of VM on the momentum p of the electron (via �Le ) [1]. After the

Fourier transform, “∇×” becomes essentially “−iQ×” (cf. [1]) and we obtain

〈�f n1 | �n · (Q × (Q × �tot

e (Q))) | �in0〉 = 〈 �f n1 | − �n · (

Q × (�tote (Q) × Q)

)︸ ︷︷ ︸ | �in0 〉

= 〈�f n1 | − �n · �tote ⊥(Q) | �in0〉

∝ 〈 �f n1 | − � · �tote ⊥(Q) | �in0〉.

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For the entire sample we have

∑�tot

e ⊥(Q) = M⊥(Q).

This leads to the magnetic cross section

d2�

d�dEf

= kf

ki

( m

2��2

)2 ∑�i , �f

n0, n1

p(�i)p(n0)∣∣〈kf �f n1|VM |ki�in0〉

∣∣2�(ε1 − ε0 − ��)

= kf

ki

(�r0)2∑

�i , �f

n0, n1

p(�i)p(n0)∣∣〈�f n1|� · M⊥(Q)|�in0〉

∣∣2�(ε1 − ε0 − ��)

and to the “magnetic selection rule”

M⊥(Q) = Q × (M(Q) × Q)

where we have the Fourier transform of the magnetization (operator) density

M(Q) =∫

d3r eiQr M(r).

Note that since M(Q) is an operator acting on the sample states, the matrix element of M(Q) forn1 = n0 (� = 0) corresponds to the Fourier transformed static magnetization density, while for n1 �= n0

(� �= 0), it may e.g. correspond to the eigenvector of a spin-wave excitation.For unpolarized incoming neutrons and no polarization analysis the sum over the neutron spin

states is straightforward. We also sum over the (complete set of) final sample energy states, rewrite the�-function as integral and write the magnetization operators in the Heisenberg picture. The magneticcross section then reads

d2�

d�dEf

= kf

ki

(�r0)2∑

�i ,�f ,n0,n1

p(�i)p(n0)∣∣〈�f n1|� · M⊥(Q)|�in0〉

∣∣2�(ε1 − ε0 − ��)

= kf

ki

(�r0)2∑n0

p(n0)1

2��

∫ ∞

−∞dt e−i�t 〈n0| M∗

⊥(Q, 0) · M⊥(Q, t) | n0〉∣∣∣∣∣︸ ︷︷ ︸

= kf

ki

(�r0)2 〈M∗⊥(Q) · M⊥(Q)〉T ,�

∣∣∣∣∣︸ ︷︷ ︸scattering function S(Q, �)

= kf

ki

(�r0)2

︷ ︸︸ ︷∑�

(�� − Q�Q) 〈M�∗(Q)M(Q)〉T ,�

∣∣∣∣∣︸ ︷︷ ︸S�(Q, �)

where we have defined a scattering function, that we will now consider in more detail.

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Figure 2. Magnetic pair correlation function G(Q, t) = 〈M�(−Q, 0)M(Q, t)〉T split into a time-independent part(blue) and “the rest” (space between blue and black curves). Only the real part is shown.

2. SCATTERING FUNCTION AND SUSCEPTIBILITY

First we note that the scattering function is the space and time Fourier transformed time dependentmagnetic pair correlation function:

S�(Q, �) = 〈M�∗(Q)M(Q)〉T ,�

= 1

2��

∫dt e−i�t 〈M�(−Q, 0)M(Q, t)〉T

= 1

2��

∫d3r dt ei(Qr−�t) 〈M�(0, 0)M(r , t)〉T︸ ︷︷ ︸ .

time dependent magnetic pair correlation function G(r , t)

Here, 〈. . .〉T ,� denotes the quantum mechanical and thermal average at a constant temperature T for agiven constant energy transfer ��, and 〈. . .〉T the thermal average at constant temperature.

Next we split the (space-)Fourier transformed pair correlation function G(Q, t) =〈M�(−Q, 0)M(Q, t)〉T (black) into a time independent part (blue) and “the rest” (red), as sketchedbelow. In this chapter the color blue marks a time-independent (static) quantity, and red a time-dependentquantity without the static part.

2.1 t → ∞ limit of pair correlations

The time independent part of the pair correlations is obtained by taking the limit t → ∞. We considerits Fourier transform:

S�static(Q, �) = 1

2��

∫dt e−i�t lim

t→∞〈M�(−Q, 0)M(Q, t)〉T

= 1

2��

∫dt e−i�t 〈M�(−Q, 0)M(Q, ∞)〉T .

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Since there are no correlated deviations from average at t → ∞:

S�static(Q, �) = 1

2��

∫dt e−i�t 〈M�(−Q, 0)〉T 〈M(Q, ∞)〉T .

The thermal average of 〈M�(Q, t)〉T is the same at t → ∞ as at t = 0:

S�static(Q, �) = 1

2��

∫dt e−i�t 〈M�(−Q, 0)〉T 〈M(Q, 0)〉T

= �(�) 〈M� ∗(Q, 0)〉T 〈M(Q, 0)〉T

which gives the magnetic Bragg peak scattering at energy �� = 0, but also the purely elastic scatteringin case of short range order like e.g. in spin glasses.

2.2 Equal-time pair correlations: t = 0

Another important limit are the equal-time pair correlations, where t = 0 in the pair correlation function.Their space Fourier transform is the energy integrated scattering function, which is called structurefactor.

〈M�(−Q, 0)M(Q, t = 0)〉 =∫ +∞

−∞d� e−i�0 S�(Q, �)

= S�(Q, t = 0) = S�(Q).

Experimentally, S(Q) can be measured by explicit measurement of S(Q, �) at constant-Q andsubsequent energy integration, e.g. on a time-of-flight (TOF) or triple-axis spectrometer (TAS). Theintegration interval needs not to be [−∞, +∞], it is sufficient to cover the full energy range of themagnetic dynamics.

On diffraction instruments, where the final neutron energy is not analyzed (energy integration) and

S(Q) =∫ +∞

−∞d� S(Q, �)

is approximated by

∫ Ei+kB T

0dEf

d�

d�dEf

=∫ Ei

−kB T

d�d�

d� dEf

∝∫ Ei

−kB T

d� S(, �, �).

This is valid if Ei � εn, ∀n for all excited magnetic states of the sample εn. The incident neutron energydefines the upper integration limit (the neutron cannot give more energy to the sample than it has itself),the lower limit is approximately given by the temperature of the sample (the neutron cannot gain moreenergy from the sample than about kBT , since the sample will not have any states thermally occupiedabove kBT ), and the scattering function S(Q, �) is zero for negative energies below kBT as well asabove all excited states. Large enough ki (Ei) is important for yet another reason: Only for Ei � εn∀n,a fixed sample orientation () and fixed kf -direction (�) with variable length of kf correspond toapproximately the same Q, otherwise each final energy (and length of kf ) corresponds to another totalmomentum transfer Q, see sketch below.

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Figure 3. Scattering triangles for a fixed sample orientation and a fixed kf -direction � at constant but too smallincident neutron wave vector ki : Different lengths of kf (blue solid and dashed green) correspond to differentmomentum transfers Q (blue solid and dashed green).

2.3 Bragg and “diffuse” scattering

We now consider what remains in the pair correlations in addition to the t → ∞ limit:

〈M� ∗(Q, 0)M(Q, t)〉T

= 〈M� ∗(Q, 0)〉 〈M(Q, 0)〉 . . .}→ Bragg, short range order, glassy frozen states: time independent

. . . + 〈(M� ∗(Q, 0) − 〈M� ∗(Q, 0)〉)(M(Q, t) − 〈M(Q, 0)〉)〉T︸ ︷︷ ︸correlation of fluctuations

and we find that after separation of all time-independent parts (Bragg, i.e. periodic long range order,short range order, even disordered frozen spin states), the “rest” corresponds to the correlations ofthe fluctuations (spin-waves and other magnetic excitations, paramagnetic scattering etc.). We call theFourier transform of the latter “diffuse”, it may nevertheless contain sharp delta-shaped excitations(cf. sketch below), and it may also be non-zero at � = 0. The sketch below illustrates the relationbetween the time-independent and the diffuse (“rest”) scattering, and their Fourier transforms which wecall Sstatic(Q, �) and Sdiff (Q, �), keeping in mind that “static” contains all time-independent scattering,even short-range order and glassy frozen states. (In some textbooks, contrary to here, all scattering whichdoes not lead to Bragg peaks is called “diffuse”, in others the entire time-independent part may be referedto as “Bragg” scattering.)

2.4 Dynamic susceptibility

The dynamic susceptibility is the linear response to a space and time varying magnetic field, defined viathe expectation value of the magnetization density (cf. [3]):

〈M�(r , t)〉T = 〈M�(r , t)〉T

∣∣H=0 +

∑

∫d3r ′ dt ′ ��(r − r ′, t − t ′)H (r ′, t ′)

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Sαβ(Q, ω) = Sαβstatic(Q, ω = 0) + Sαβ

diff (Q, ω)

Figure 4. Sketch of the time-independent and time-dependent pair correlations (here shown as real), and theirFourier transforms, the static and “diffuse” scattering functions. Two different examples of diffuse scattering aredisplayed.

Fourier transform

〈M�(Q, �)〉T = 〈M�(Q, �)〉T

∣∣H=0 +

∑

��(Q, �)H (Q, �)

〈M�(Q, �)〉T

∣∣H=0 contains the Fourier transform of the field independent magnetization density, it will

be a �-peak at zero energy (dynamic contributions cancel each other in the ensemble average). Before welook closer at the dynamic susceptibility, we consider which susceptibility we measure with a SQUID.The SQUID measures with a field at zero frequency (static), the magnetization is integrated over timeand all the sample space: ∫

d3r dt 〈M�(r , t)〉T =∫

d3r dt ei(0t−0r)〈M�(r , t)〉T

= 〈M�(Q = 0, � = 0)〉T

= �′ ��(Q = 0, � = 0)H �

since �′′ ��(Q = 0, � = 0) = 0 = ���(Q = 0, � = 0)H �

and hence correponds to the dynamic susceptibility at Q = 0, � = 0.For general Q, �, the Fourier transformed ��(Q, �) = �′�(Q, �) + i�′′�(Q, �) is complex, and in

general ��(Q, �) is not the same as ��∗(Q, �), i.e. the dynamic susceptibility tensor is not a Hermitianmatrix.

2.5 Diffuse scattering function and dynamic susceptibility: The fluctuation-dissipationtheorem

The “diffuse” part of the scattering function is closely related to the dynamic susceptibility. One canshow [1, 3] that

S�diff (Q, �) = S�(Q, �) − S

�static(Q, �)

= −��

1

1 − e− ��

kB T

1

2i(��(Q, �) − �� ∗(Q, �))

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S��diff (Q, �) = −�

�

1

1 − e− ��

kB T

�′′ ��(Q, �)

Sdiff (Q, �) = −��

1

1 − e− ��

kB T

·∑�

(�� − Q�Q

) 1

2

(�′′ �(Q, �) + �′′ �(Q, �)

).

These relations are valid independent of the symmetry of the sample. They relate the scattering dueto fluctuations to the imaginary part of the susceptibility, and are known as the fluctuation-dissipationtheorem.

The second and third relation describe measurable quantities and are real, while the first may becomplex. (All observables contain of course real combinations of S

�diff (Q, �).)

2.6 Kramers-Kronig relation

The Kramers-Kronig relations relate the real part of the complex susceptibility to the imaginary part.An imaginary part determined by neutron scattering allows to reconstruct the whole susceptibility:

�′ ��(Q, �) = 1

�

∫d�′ �′′ ��(Q, �′)

�′ − �·

In the context of critical scattering the so-called staggered susceptibility is important, which measuresthe linear answer of a compound at a non-zero wave vector Q, and which cannot be measured with aSQUID. Neutron scattering in a finite energy interval and subsequent energy integration allows to obtainthis staggered susceptibility via the Kramers-Kronig relation:

�′ ��(Q, � = 0) = 1

�

∫ ∞

−∞d�

�′′��(Q, �)

� − 0= −

∫ ∞

−∞d�

1 − e− ��

kB T

��S��diff (Q, �)

≈ −∫ �cutoff

−kB T

d�1 − e

− ��kB T

��S��diff (Q, �)

where again the limits −∞, ∞ can be approximated by finite limits, since S��diff (Q, �) becomes zero

below −kBT and above all magnetic excitation energies εn of the sample.The Kramers-Kronig relations also allow to relate the neutron response to the SQUID-susceptibility

�′ ��(Q = 0, � = 0):

�′ ��(Q = 0, � = 0) ≈ −∫ �cutoff

−kB T

d�1 − e

− ��kB T

��S��diff (Q = 0, �).

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2.7 Detailed balance

Generally valid are the following relations for the dynamic susceptibility and the diffuse scatteringfunction [3]:

��(Q, �) = �� ∗(−Q, −�)

S�diff (Q, �) = S

� ∗diff (Q, �)

S�diff (Q, �) = e

��kB T S

�diff (−Q, −�),

Sdiff (Q, �) = e��kB T Sdiff (−Q, −�)︸ ︷︷ ︸

Detailed balance

where the last equation is obtained by summation∑

�

(�� − Q�Q

)on both sides.

In case of centrosymmetry, i.e. for crystals with centrosymmetric structures and for isotropic media(liquids, powders), the following relations hold [3]:

�′ �(Q, �) = �′ �(Q, −�)

�′′ �(Q, �) = −�′′ �(Q, −�)

S�diff (Q, �) = e

��kB T S

�diff (+Q, −�),

Sdiff (Q, �) = e��kB T Sdiff (Q, −�)︸ ︷︷ ︸

Detailed balance for centrosymmetry

where again the last equation is obtained by summation∑

�(�� − Q�Q).One and the same linear answer of the magnetic material will therefore at different temperatures

lead to different scattering functions. Note that the scattering intensity is very different at negative andpositive energy transfer at low temperatures – a neutron cannot gain energy from the sample if thereare no thermally excited states present, but it can always, even at T = 0, give energy to the sample andcreate an excited state.

If one is interested in the temperature dependent changes of the compound’s magnetic behaviour,one should always look at the dynamic susceptibility rather than the scattering function, since S(Q, �)also depends on the changing occupation number of the excited states. A frequent question addressed byneutron scattering concerns the opening of a gap as the temperature is lowered. S(Q, �) at low energieswill generally decrease with decreasing T because of the changing occupation number, so that S(Q, �)taken at different temperatures quite frequently looks as if there were a gap opening at low temperatures.Only an analysis of the dynamic susceptibility as a function of temperature will correctly answer thequestion. In the example below, the dynamic susceptibility is temperature independent and gapless (finiteat every � > 0). There is therefore no gap, neither at low (blue) nor high (red) temperatures. If analysedin terms of a damped harmonic oscillator, the characteristic frequency of the oscillator is zero at bothtemperatures.

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Figure 5. S(Q, �) at different temperatures T for a temperature independent dynamic susceptibility �′′(Q, �),(increasing T is shown as black-blue-cyan-green-red).

S(Q, ω) χ′′(Q, ω)

Figure 6. Is there a gap or not at low temperature? – There is no gap! The “shift” of the maximum in S(Q, �) fromlow (blue) to high temperatures (red) can be misleading: The dynamic susceptibility, here chosen as temperatureindependent, is gapless and finite at every infinitesimally small � > 0, and the characteristic frequency of theharmonic oscillator is zero.

3. CRYSTALS: “SPIN-ONLY” AND SPIN-LATTICE APPROXIMATIONS

Important approximations allow to rewrite the neutron scattering cross section not only as Fouriertransformed pair correlation of the space and time-dependent magnetization density, but directly asFourier transform (FT) of the spin pair correlation function, multiplied with form factor and Debye-Waller factor. We first review the conditions of the approximations and then some generalisations, whichmake this approximation so successful, that, in practice, it is always used [1, 2, 4].

3.1 Ionic crystals with spin-only moment

We suppose now

- a ionic crystal where the unpaired electrons are localized around the equilibrium position of thenucleus Rn(t) (Heitler-London model)

- LS coupling: S = ∑ν sν and L = ∑

ν �ν

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- spin-only magnetic moment: L = 0- that the neutron energy is too small to change S or L or 〈L〉.

This means, that the unpaired electrons will have the same spatial wave functions and the same totalspin S before and after the scattering process, only the total spin direction S(t)/S and the position of theion Rn(t) (which is also supposed to be the mean position of the unpaired electrons) can change. Thismeans we will average as follows (written very sloppily, see [1] for a thorough treatment):

For a given ion n at Rn(t)we have an unpaired electron ν at r(t) = Rn(t) + rν(t)

and somewhat more precisely (cf. [1] for a proper derivation) we have for each unpaired electronspin sν

sν(r(t), t) = sν(Rn(t) + rν(t), t)

= sν(t)�(r − (Rn(t) + rν(t)))

= sν(t)∫

d3r ′ �(r − Rn(t) − r ′)�(r ′ − rν(t))

≈ sν(t)∫

d3r ′ �(r − Rn(t) − r ′)∫

d3r ′′�(r ′ − r ′′)fn(r ′′)

= sν(t)∫

d3r ′ �(r − Rn(t) − r ′)fn(r ′)

where fn(r) is the probablity to find the unpaired electron at distance r from the ion nucleus, and onlydepends on the ion n but not on the individual electron ν nor on time. We have then

∑ν

sν(r(t), t) ≈∑

ν

sν(t)∫

d3r ′ �(r − Rn(t) − r ′)fn(r ′)

= S(t)∫

d3r ′ �(r − Rn(t) − r ′)fn(r ′)

= S(Rn(t), r , t)

where S(Rn(t), r , t) · S(Rn(t), r , t) = S(S + 1) is time independent (S may be different on differentsites n).

We can therefore write

〈n1kf | M(r) | n0ki〉 ∝ 〈n1kf |∑

ν

sν(r) | n0ki〉

≈ 〈n1kf | S(Rn, r) | n0ki〉contract kf , ki and take the Fourier transform, the convolution integral of the spin positions gives aproduct of Fourier transforms

〈n1kf | M(r) | n0ki〉 = 〈n1| eiQRnfn(Q)S | n0〉and in the cross section, the time-independent fn(Q) can be taken out of the matrix element:

|〈n1kf | M(r) | n0ki〉|2�(ε1 − ε0 − ��)

∝∫ ∞

−∞d� e−i�t 〈n0|e−iQRn(0)fn(Q)∗S(0) eiQRn(t)fn(Q)S(t) | n0〉

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= |fn(Q)|2∫ ∞

−∞d� e−i�t 〈n0|e−iQRn(0)S(0) eiQRn(t)S(t) | n0〉

fn(Q) =∫

d3r f (r)eiQr

is called the magnetic form factor, it is the space-Fourier transform of the unpaired electron shell’sspatial wave function. Therefore, for not too large neutron energies we can use the ionic form factoreven in the inelastic case.

If we summarize this into a simple picture, the approximations correspond to an unpaired electronshell which is firmly fixed to the nucleus, and where the unpaired electrons are smeared out andcompletely delocalized, so that their positions are only given by a probability function. In the neutronscattering process the position of the ion can change (but the unpaired electron shell remains firmlyattached to it and does not change shape), and the direction of the total electron spin can change, but notits length S.

3.2 Spin only – “extended validity”

For ionic crystals, even if L �= 0, the Fourier transform of the unpaired electron shell can still be takenoutside of the matrix element, under the following conditions:

- quenched orbital momentum: 〈L�〉 = 0 for � = x, y, z

- LS-coupling with only one value of J : J = S + L, EJ � EJ ′

- neutron energy small

compared to higher < L >-, L- or J -states: �� � EJ ′ , E<L>′

- small Q: |Q|−1 � extension of

|Q|−1 � unpaired electron shell.The table below summarizes how f (Q) and S should be replaced in various cases.

Case Condition (in addition to 3.1) example

∣∣∣∣true L = 0 – f (Q) S Mn2+

∣∣∣∣〈L〉 = 0 small Q g

2 f (Q) S Cu2+∣∣∣∣

J = S + L small Q, EJ � EJ ′ �=JgJ

2 f (Q) J most but not Sm3+, Eu2+∣∣∣∣

The transition matrix elements become then

〈n1kf | M(r) | n0ki〉 =∑

n

gn

2fn(Q)〈n1| eiQRnJ (Rn) | n0〉

〈n1kf | M(r) | n0ki〉 =∑

n

gn

2fn(Q)〈n1| eiQRnS(Rn) | n0〉.

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Figure 7. Form factor of the Cu2+ ion, after [6].

If in addition the spatial electron wave function can be approximated as spherically symmetric, the formfactor depends on Q = |Q| only:

gJ

2f (Q) ≈ gS

2J0(|Q|) + gL

2

( J0(|Q|) + J2(|Q|) )

Jn(|Q|) = 4�∫ ∞

0dr r2 jn(|Q|r)︸ ︷︷ ︸s(r)

spherical Bessel function of order n

This approximation, together with the small-|Q| condition, is called the dipolar approximation (cf. [6]).The form factor of the Cu2+ ion [6] is shown above.

3.3 Spin-lattice decoupling approximation

In addition to spin-only and dipolar approximation we will now assume that the electron shell spindirection has no effect on the position of the ionic nucleus nor the mean position of the unpaired spinelectron cloud (remember that both are firmly coupled in the Heitler-London model). This assumptionmay not be valid for every aspect in e.g. spin-Peierls compounds and multiferroics. We further assumethat the mean position of the unpaired spin electron cloud is the equilibrium position of the ion, and nolonger t-dependent.

We can then separate the correlation function of the ion positions from the correlation of spinorientations

〈e−iQRn(0)S�(Rn(0), 0) eiQRm(t)S(Rm(t), t)〉T ≈ 〈e−iQRn(0)eiQRm(t)〉T 〈S�(Rn, 0)S(Rm, t)〉T

and write

Rn(t) = Rn︸︷︷︸ + un(t)︸︷︷︸equilibrium position displacement.

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For simplicity we now also assume a Bravais lattice. This makes the formula easier to read, but is notcrucial for the approximation.

∑nm

〈e−iQRn(0) S�(Rn(0), 0) eiQRm(t) S(Rm(t), t)〉T

≈∑

n

eiQRn

∫dt e−i�t 〈e−iQu(0)eiQun(t)〉T

∣∣∣︸ ︷︷ ︸∣∣∣〈S�(0, 0)S(Rn, t)〉T︸ ︷︷ ︸

=∑

n

eiQRn

∫dt e−i�t (GN (n, ∞) + GN (n, t)) (G�

M (n, ∞) + G�M (n, t)).

At this stage several text books introduce four different types of scattering, according to thefour terms that can be formed of the product of the time-independent term and the fluctuationterm of the pair correlation functions. GN (n, t)G�

M (n, ∞) is typically called magnetovibrationalscattering, and GN (n, t)G�

M (n, t) is called the fully inelastic term. Note, however, that theseterms cannot describe electromagnons nor any kind of correlated spin-lattice excitation, sincewe arrived at this stage assuming that spin directions and ion positions are uncorrelated.I am not aware of a measurement of these terms, we will simply neglect them in the following. Wethen remain with

≈∑

n

eiQRn

∫dt e−i�t GN (n, ∞) (G�

M (n, ∞) + G�M (n, t))

= 〈e−iQu〉T 〈eiQu〉T

∑n

eiQRn

∫dt e−i�t (G�

M (n, ∞) + G�M (n, t))

︸ ︷︷ ︸= e−2W (Q) 〈S� ∗(Q)S(Q)〉T ,�

where we have the Debye-Waller factor multiplied with the spin pair correlation function (static anddiffuse part). We can imagine the Debye-Waller factor roughly as the Fourier transform of the time-averaged smeared-out ion position. It decreases with increasing |Q| and increasing T and is essentially1 in the limit |Q| → 0, T → 0, except for quantum crystals and quantum liquids (like Helium).

3.4 Unpolarized magnetic neutron cross section

For a Bravais lattice, the cross section with the previous assumptions reads now:

magnetic “scattering length”

d2�

d dEf

= (�r0)2

2��

kf

ki

∑�

(�� − Q�Q)(g

2f (Q)

)2e−2W (Q) 〈S� ∗(Q)S(Q)〉T ,�︸ ︷︷ ︸

geometric magnetic Debye-Waller S�static(Q, � = 0) + S

�diff (Q, �)

selection rule form factor factor scattering function

where S�(Q, �)( g

2 f (Q))2 e−2W (Q) = S�(Q, �), i.e. form factor and Debye-Waller factor have beentaken out of S�(Q, �).

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3.5 Sum rules

Important and useful in practice are a number of relations for the zeroth, first and second moment ofthe scattering function with respect to energy which are known as (frequency or f-) sum rules. Here weconsider the zeroth moment using the example of a Bravais ferromagnet in a single domain state withmagnetization parallel to z.

In a neutron scattering experiment we will find static Bragg as well as “diffuse” scattering, the lattercontains magnon creation as well as magnon annihilation terms.

Szzstatic(Q, �) = N〈Sz〉2

T(2�)3

v0�(�)

∑� �(Q − �)

Syy

diff (Q, �) = S2

(2�)3

v0

∑�,q〈1 + nq〉T �(Q − q − �) �(�� − ��q) . . . magnon creation

. . . + 〈nq〉T �(Q + q − �) �(�� + ��q) magnon annihilation

��q = 2SJ (1 − cos(qaa)) magnon dispersion

nq = e− ��q

kB T

1−e− ��q

kB T

= 1

e

��qkB T −1

→ 0 for T → 0 Bose factor

Here � is a reciprocal lattice vector and v0 the volume of the elementary cell.We notice that the Bragg peak intensity is N〈Sz〉2

T , while the magnon intensity is S2 for S��(Q, �)

with a direction � = x, y orthogonal to the magnetically ordered moment. The Bragg peak is located atthe Brillouin zone center, while the magnon exists at all N q-points in the Brillouin zone. For the totalintegral over the scattering function, integrated over energy and the first Brillouin zone we obtain:

∫d3Q d�

(Szz

static(Q, �) + Syy

diff (Q, �) + Sxxdiff (Q, �)

)= N

∑�

〈( S�)2〉T

= NS(S + 1)

for T → 0 = NS2 + NS.

For T → 0, in a classical ferromagnet, the Bragg peak will hence contain NS2, the inelastic scattering(the scattering from magnons) NS. As the temperature increases, more and more magnons will becreated as well as annihilated in the neutron scattering process, the inelastic intensity will grow, andintensity will be missed in the Bragg peak, while the total sum remains constant. Note that the totalsum is independent of the interactions, and therefore a general rule, not only valid for ferromagnets,but also antiferromagnets, short-range ordered and disordered spin systems, paramagnets, and quantumspin magnets that do not order for T → 0, where hence all scattering is diffuse (inelastic).

4. POLARIZED NEUTRONS

This section summarizes the essentials of polarized neutron scattering for inelastic single crystalscattering. Specialized cases (energy integrating or isotropic scattering) are dealt with in other chapters,and more detailed courses can be found in the lecture book of a previous JDN-school devoted topolarized neutron scattering [10]. The neutron spin properties are marked red, sample propertiesaffecting the neutron spin are blue, and sample properties that do not affect the neutron spin are green

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in this section. The cross section for a specific transition of the neutron spin state �i → �f is

d2�

d� dEf

∣∣∣�i→�f

= kf

ki

∑n0

p(n0)∣∣ 〈n1�f | V (Q) | n0�i〉

∣∣2�(ε1 − ε0 − ��)

where V (Q) =∑

n

eiQRn(

An + Bn � · I︸︷︷︸ ) −�r0 � · M⊥(Q)︸ ︷︷ ︸is the combined scattering potential. The interaction with the neutron spin has the same form (scalarproduct) for the nuclear and the electronic spin, and we combine them into the symbolM

〈�f n1 | � ·M | n0�i〉 =∑

�

〈�f n1 | ��M� | n0�i〉

≈∑

�

〈�f |��|�i〉∣∣︸ ︷︷ ︸

∣∣〈n1|M�|n0〉︸ ︷︷ ︸ .

Dipole matrix elements

For completely uncorrelated nuclear spins we can separate the matrix elements for the neutron spin andthe sample, which is extremely useful for e.g. crystal field transitions. We thus obtain the standard dipoletransition matrix elements (second line).

Since �z leaves its own eigenstates unchanged, but �x,y changes �z = +1 to �z = −1 and vice versa,i.e. flips the neutron spin from +z to −z, a useful rule to memorize is

- the neutron spin is not changed by magnetic moments or magnetic eigenvectors (fluctuationamplitudes) that are parallel or antiparallel to it

- the neutron spin is flipped by magnetic moments or magnetic eigenvectors (fluctuationamplitudes) that are orthogonal to it.

Note that this rule is not valid for correlated nuclear spin moments or amplitudes, as in this case thematrix elements for neutron and nuclear spin cannot be separated.

4.1 Blume-Maleyev equations

For uncorrelated nuclear spin directions and uncorrelated nuclear isotope distribution, that are notcorrelated with the electronic magnetization density, the general cross section for polarized neutronsscattering from nuclei and unpaired electrons is written as follows (see [7, 8] for the equal-timecorrelations and [9] for the inelastic scattering in this form):

d2�

d� dEf

= 〈N∗N〉T ,� + 〈M∗⊥M⊥〉T ,� + 〈N∗[P i · M⊥]〉T ,� + 〈N [P i · M∗

⊥]〉T ,�

+iP i · 〈[M∗⊥ × M⊥]〉T ,� + 〈νi〉T + 〈�nsi〉T

Pf

d2�

d� dEf

= P i

( 〈N∗N〉T ,� + 〈νi〉T

) −P i〈M∗⊥M⊥〉T ,� − 1

3P i〈�nsi〉T

+〈M⊥[P i · M∗⊥] + M∗

⊥[P i · M⊥]〉T ,� − i 〈M∗⊥ × M⊥〉T ,�

+〈N∗M⊥ + NM∗⊥〉T ,� − i 〈N∗[P i × M⊥] + i N [P i × M∗

⊥]〉T ,�

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where we have abbreviated

nuclear coherent scattering N = N (Q)electronic magnetic scattering M⊥ = M⊥(Q)nuclear spin incoherent scattering �nsi

nuclear isotope incoherent νi

incoming neutron spin polarization P i

outgoing neutron spin polarization Pf

time Fourier transform and thermal average 〈. . .〉T ,� = 12�

∫ ∞−∞ dt e−i�t 〈. . .〉T .

For inelastic scattering (with its comparably small intensities), it is practical to consider directly the crosssections for a given incident and outgoing neutron spin state. For diffraction, it can be more practicalto consider the polarisation of the outcoming beam as a function of the incoming neutron polarisation(polarisation matrix). We will now go through several common experimental situations for inelasticscattering.

For all of these we define an orthogonal, right-handed coordinate system, with x = Q and Q =ki − kf , z perpendicular to the scattering plane, and y = z × x.

4.2 Magnetic field in the scattering plane

In a magnetic field along the scattering wave vector (H ||Q), with incoming unpolarized neutrons, thepolarisation of the scattered neutrons can be analyzed parallel (x) and antiparallel (x) to H ||Q||x, andthe two corresponding cross sections are:

d2�

d� dEf

∣∣0→x

= 1

2[〈N∗N〉T ,� + 〈M∗

⊥M⊥〉T ,� + 〈νi〉T + 〈�nsi〉T ]− i

2〈[M∗

⊥ × M⊥]〉T ,� · x

d2�

d� dEf

∣∣0→x

= 1

2[〈N∗N〉T ,� + 〈M∗

⊥M⊥〉T ,� + 〈νi〉T + 〈�nsi〉T ]+ i

2〈[M∗

⊥ × M⊥]〉T ,� · x.

Subtraction of the two cross sections allows to eliminate background and nuclear scattering and to isolatethe chiral term i〈[M∗

⊥ × M⊥]〉T ,� · x. The chiral term occurs in form of Bragg peaks in cycloid andhelical structures, but also in inelastic neutron scattering, not only for excitations from chiral magneticstructures, but already in ferromagnets. The precession of the spins around the ordering directionx||H ||Q in a ferromagnet can be described by an eigenvector M⊥ ∝ (01i). Its vector product withthe complex conjugate is as large as the scalar product M∗

⊥ · M⊥, so that only one of the two crosssections shows the magnetic excitation. In ferromagnets, where acoustic phonons and magnons emergeboth at reciprocal lattice points Q = �, subtraction of these two cross sections allows to isolate the purechiral signal of the magnetic excitations. An example is shown below [11].

If the incoming neutrons are polarized parallel (x) or antiparallel (x) to H ||Q||x, and theirpolarization is analysed as well, the corresponding four cross sections are

d2�

d� dEf

∣∣x→x

= d2�

d� dEf

∣∣x→x

= 〈N∗N〉T ,� + 〈νi〉T + 1

3〈�nsi〉T

d2�

d� dEf

∣∣x→x

= 〈M∗⊥M⊥〉T ,� + 2

3〈�nsi〉T +i〈[M∗

⊥ × M⊥]〉T ,� · x

d2�

d� dEf

∣∣x→x

= 〈M∗⊥M⊥〉T ,� + 2

3〈�nsi〉T −i〈[M∗

⊥ × M⊥]〉T ,� · x.

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Figure 8. Inelastic chiral term of magnons in a ferromagnet, emerging from a nuclear Bragg peak, measured ina horizontal magnetic field with H ||Q. The left side shows the chiral term measured as difference of the twocross sections with incoming unpolarized neutrons, the intensity is color-coded. The right side shows the four crosssections for incoming polarized neutrons and polarisation analysis for one constant-energy scan. The magnon canbe seen as a peak in only one of the four cross sections. Measured on IN20, taken from [11].

Figure 9. Inelastic chiral term of a mono-domain screw-type magnetic structure (langasite), here measured via12 (�xx − �xx) on IN20, ILL. The intensity is color-coded. Taken from [12].

The inelastic chiral term has also been measured in a langasite, which has a crystal structure withoutinversion center and orders into a screw-type structure with only one chiral domain [12], see above.

4.3 Magnetic field perpendicular to the scattering plane

With polarized incoming neutrons, polarization analysis after the sample and a magnetic fieldperpendicular to the scattering plane, the initial and final neutron polarization is prepared and measuredalong the field axis, parallel or antiparallel to the field, P ||H ||z. Four cross sections can be measured:

d2�

d� dEf

∣∣↑↑ = 〈N∗N〉T ,� + 〈M∗

⊥ zM⊥ z〉T ,� + 〈νi〉T + 1

3〈�nsi〉T +〈[N∗M⊥ z + NM∗

⊥ z]〉T ,�

d2�

d� dEf

∣∣↓↓ = 〈N∗N〉T ,� + 〈M∗

⊥ zM⊥ z〉T ,� + 〈νi〉T + 1

3〈�nsi〉T −〈[N∗M⊥ z + NM∗

⊥ z]〉T ,�

d2�

d� dEf

∣∣↑↓ = 〈M∗

⊥ yM⊥ y〉T ,� + 2

3〈�nsi〉T

d2�

d� dEf

∣∣↓↑ = 〈M∗

⊥ yM⊥ y〉T ,� + 2

3〈�nsi〉T .

This setup is useful for those inelastic cases, where the anisotropy does not allow to keep Q||H forthe Q-vectors that one wishes to measure, or if the direction of the applied field matters (e.g. different

01002-p.20

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magnetic phases for different field directions with respect to the crystal axes), or if one wishes to separatethe magnetic components parallel and perpendicular to the magnetic field. In absence of phonons at theQ, � in question, this works generally well, even though a full isolation of the magnetic signal from theincoherent scattering and background is not possible.

For diffraction, the most important application consists in measuring the nuclear-magneticinterference which is directly obtained as the difference of the ↑↑ and ↓↓ cross sections. Since theother two cross sections are identical, it is not necessary to analyse the outgoing polarisation:

d2�

d� dEf

∣∣↑→0 = 〈N∗N〉T ,� + 〈M∗

⊥ · M⊥〉T ,� + 〈νi〉T + 〈�nsi〉T +〈[N∗M⊥ z + NM∗⊥ z]〉T ,�

d2�

d� dEf

∣∣↓→0 = 〈N∗N〉T ,� + 〈M∗

⊥ · M⊥〉T ,� + 〈νi〉T + 〈�nsi〉T −〈[N∗M⊥ z + NM∗⊥ z]〉T ,�

and the interference term is obtained as the difference of the two cross sections with opposite incomingpolarisation.

4.4 Polarization analysis along three perpendicular directions

A very powerful method consists in using polarized incident neutrons and polarisation analysis afterscattering at the sample, and measuring the cross sections for three orthogonal axes of the neutronpolarisation. The axis of the neutron polarization at the sample can be defined by a weak magneticfield (1-6 mT) which for many cases will be equivalent to zero field (Helmholtz setup), or alternativelybe prepared field-free (Cryopad, Mupad). This method is sometimes called longitudinal polarizationanalysis or diagonal polarization analysis in order to indicate that the quantization axis for the neutronspin is the same for the incident and the scattered beam. If we define as before a righthanded spincoordinate system x||Q = ki − kf , z vertical, y = z × x, and abbreviate the differential cross section

d2�d� dEf

∣∣� as ��, we have the following six equations

�xx = �xx = 〈N∗N〉T ,� + 〈νi〉T + 13 〈�nsi〉T

12�xx + 1

2�xx = +〈M∗⊥ yM⊥ y〉T ,� +〈M∗

⊥ zM⊥ z〉T ,� + 23 〈�nsi〉T

12�yy + 1

2�yy = 〈N∗N〉T ,� + 〈νi〉T +〈M∗⊥ yM⊥ y〉T ,� + 1

3 〈�nsi〉T

�yy = �yy = +〈M∗⊥ zM⊥ z〉T ,� + 2

3 〈�nsi〉T

12�zz + 1

2�zz = 〈N∗N〉T ,� + 〈νi〉T +〈M∗⊥ zM⊥ z〉T ,� + 1

3 〈�nsi〉T

�zz = �zz = +〈M∗⊥ yM⊥ y〉T ,� + 2

3 〈�nsi〉T

We can now isolate the purely magnetic scattering either from the “non-spin flip” cross sections, wherethe neutron spin remains unchanged in the scattering process, or from the “spin-flip” cross sections,where the neutron spin is reversed in the scattering process. The latter is statistically favorable inpresence of phonons in the spectrum, the former is favorable, if not phonons, but rather the (nuclearspin)- incoherent background is the dominant nuisance:

〈M∗⊥M⊥〉T ,� = �xx + �xx −�yy − �zz

〈M∗⊥M⊥〉T ,� = 1

2�yy + 12�yy + 1

2�zz + 12�zz −2�xx .

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Figure 10. Measurement of the rotation sense of a cycloid magnetic structure in an electric field. The figures displaythe cross sections �yx (open red circles) and �yx (filled blue circles) measured on IN20 with orthogonal incoming(||y) and outgoing neutron polarization (|| ± x). Taken from [13].

We can also separate different directions of the magnetic moments or fluctuation amplitudes(eigenvectors of excitations), the chiral term, the pure nuclear-spin incoherent scattering, or (the realpart of) the nuclear magnetic interference terms, provided the sample is in a single-domain state:

〈M∗⊥ yM⊥ y〉T ,� = 1

2�yy + 1

2�yy − �xx

= 1

2�xx + 1

2�xx − �yy

〈M∗⊥ zM⊥ z〉T ,� = 1

2�zz + 1

2�zz − �xx

= 1

2�xx + 1

2�xx − �zz

4

3〈�nsi〉T = 2�yy + 2�zz − (�xx + �xx)

2i x · 〈[M∗⊥ × M⊥]〉T ,� = �xx − �xx

2〈N∗M⊥ y + NM∗⊥ y〉T ,� = �yy − �yy

2〈N∗M⊥ z + NM∗⊥ z〉T ,� = �zz − �zz .

For weak magnetic signals, the separation of the magnetic signal from both phonons and incoherentsignals is essential. Measurement of just �xx leads to erroneous results, since the incoherent nuclearspin scattering frequently displays an energy dependence which follows the density of states of allexcitations in the sample, due to multiple scattering processes in the sample (incoherent plus inelastic).

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It is further possible to analyse the final neutron polarisation along an arbitrary direction, evenperpendicular to the initial neutron polarisation. For general Q, � this can be achieved in a well-controlled manner, if the sample is in a strict zero-field environment. Such an experimental setupallows to determine two additional correlations (the imaginary parts of the nuclear-magnetic interferenceterms), which are important for the analysis of complex magnetic structures, cf. chapter of E. Ressouche.

Below we show an example of a cycloid ferroelectric where the incoming neutron polarization wasprepared along y and the outgoing neutron polarization measured along ±x. The rotation sense of thecycloid magnetic structure was commanded by an electric field [13] and could also be switched (notshown here). In this example the chiral term has nearly the same size as the scalar product M∗

⊥M⊥ sothat for each of the three wave vectors a peak appears in only one of the two cross sections �yx , �yx .

5. CONCLUSION

Neutron scattering is a unique technique for studies of magnetism, since it probes directly the Fouriertransform of the two-spin correlation function, or more generally, the magnetic pair correlations. Dueto the weakness of the neutron-matter interaction, there is normally only one scattering process foreach neutron, and the correlation functions are probed in the whole volume. (In large samples, andfor strong scattering (Bragg peaks), as well as when searching for exceedingly small signals, multiplescattering nevertheless needs to be taken into acount.) Neutrons are sensitive to electronic magneticmoments or fluctuation amplitudes perpendicular to the momentum transfer. Elastic scattering (� = 0)reflects the t → ∞ correlations (Bragg peaks, short range order, glassy frozen states). Diffraction(energy integrated detection) is proportional to the equal-time correlations (t = 0). Inelastic scatteringreflects the correlated fluctuations, and is directly related to the dynamic susceptibility. Finally, polarisedneutrons allow to measure the magnetic signal, its individual components (and more), and to isolate themfrom other contributions as well as from background.

References

[1] G.L. Squires, Introduction to the Theory of Neutron Scattering, Dover Publications, 1996.[2] P.G. de Gennes, Theory of Neutron Scattering by Magnetic Crystals, in Magnetism, Vol. III,

Eds. G.T. Rado and H. Suhl (New York: Academic Press, 1963), p.115–147.[3] W. Marshall and R.D. Lowde, Magnetic Correlations and Neutron Scattering, Rep. Progr. Phys.

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